To approximately sample from the target \(\pi \) , one then selects a level \(L\)  such that \(\pi ^{(L)}\)  is a sufficiently accurate approximation of \(\pi \)  and then applies Monte Carlo or particle methods to \(\pi ^{(L)}\) ; see, e.g., [1]}, [2]}, [3]}.
One challenge of such an approach is that the density of \(\pi ^{(L)}\)  can be computationally expensive to evaluate, because each evaluation of the density entails at least one numerical solve of the PDEs underlying the forward model, which can quickly make sampling from it prohibitively expensive.
